In actual application scenarios, there will be errors in the measured angle, which will affect the expected user reception performance and reduce the security of wireless transmission. However, in the above research, the direction angle is assumed to be ideal known information, thus further study on systems with imperfect information of direction angle is needed. Hu et al. [1]} designed a robust DM synthesis technique based on the principle of minimum mean square error (MMSE) in view of the uncertainty of the estimated direction angle, which improves the bit error rate (BER) performance of the system by minimizing the constellation distortion in the desired direction. Shu et al. [2]} considered a multi-beam broadcast DM scenario with an imperfect desired direction. [2]} designed beamforming by minimizing information leakage in the eavesdropping direction, and maximizing the expected average received signal-to-interference-to-noise ratio (SINR) of the user end to design a projection matrix of AN, so as to achieve the robustness of CMs transmission in the wireless communication system. Shu et al. [4]} considerd the MIMO system with direction of arrival (DOA) measurement error, and proposed a robust beamforming scheme based on the combination of main lobe integration and leakage, which transmits independent and parallel private data streams to multiple legitimate users at the same time, thereby realizing the stable and safe transmission of CMs in the DM system. Shu et al. further designed a low-complexity scheme in [5]} that can accurately and safely transmit CMs to desired users through the joint use of AN, phase alignment, orthogonal frequency division multiplexing (OFDM) technology, and DM random subcarrier selection technology.
